I've figured out what is causing it. I'm using the "More Perks" mod by Dree74 with the destiny gambler perk. It's supposed to give you random effects that change by the day, which caused the skills to be depleted for a time. But I suppose the game glitched and made those permanent somehow. I've tried using the console to remove the "spell" that causes it, but the game keeps returning an error saying I don't have that "spell." I also tried removing the mod, but the effect stays. Unless someone knows a fix for this, I'll have to go back a few saves before the glitch happened. It's gone unnoticed for a while though, so I'll have to replay quite a bit of the game.
Glad you figured it out - looking on FONV Nexus for this mod it seems the issue of stats permanently setting to zero is fixed in the latest release, so make sure you have the most up to date version
If you already had the most up to date version be sure to report this problem so that the author can find a fix.
Edit to add: also changing your timescale can glitch things that are supposed to reset on a particular day or after a particular number of hours, so there may be a conflict if you've been fiddling with the timescale settings.